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gr-qc/9610066

Does matter differ from vacuum?

Christoph Schiller∗

Service de Chimie-Physique, CP 231Universite Libre de Bruxelles

Boulevard du Triomphe1050 Bruxelles, Belgium

Abstract

A structured collection of thought provoking conclusions about space and time is given.Using only the Compton wavelength λ = h/mc and the Schwarzschild radius rs =2Gm/c2, it is argued that neither the continuity of space-time nor the concepts ofspace-point, instant, or point particle have experimental backing at high energies. It isthen deduced that Lorentz, gauge, and discrete symmetries are not precisely fulfilled innature. In the same way, using a new and simple Gedankenexperiment, it is found thatat Planck energies, vacuum is fundamentally indistinguishable from radiation and frommatter. Some consequences for supersymmetry, duality, and unification are presented.

PACS numbers: 03.65.Bz, 04.20.Cv

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1 The limits of textbook physics

Matter and space-time are the fundamental entities of our description of nature. Whydo particles have the masses they have? The quantum theory of the electromagnetic,weak and strong nuclear forces cannot answer the question. Why does space-time havethree plus one dimensions? The generally accepted theory of the structure of space-time, general relativity, cannot explain this simple observation (and in fact does noteven allow to ask this question). One concludes that the description of nature by thesetheories is still incomplete.

The search for a more complete, unified description of nature is also driven byanother, even more compelling motivation: quantum mechanics and general relativitycontradict each other; they describe the same phenomena in incompatible ways. Thiswell-known fact surfaces in many different ways. [1]

There is a simple, but not well known way to state the origin of the contradictionbetween general relativity and quantum mechanics. Both theories describe motionusing objects, made of particles, and space-time. Let us see how these two conceptsare defined.

A particle – and in general any object – is defined as a conserved, localisable entitywhich can move. (The etymology of the term ‘object’ is connected to this fact.) Inother words, a particle is a small entity with conserved mass, charge etc., which canvary its position with time.

At the same time, in every physics text [6] time is defined with the help of movingobjects, usually called ‘clocks’, or with the help of moving particles, such as thoseemitted by light sources. Similarly, also the length unit is defined with objects, be it aoldfashioned ruler, or nowadays with help of the motion of light, which is a collectionof moving particles as well.

The rest of physics has sharpened the definitions of particles and space-time. Inquantum mechanics one assumes space-time given (it is included as a symmetry ofthe Hamiltonian), and one studies in all detail the properties and the motion of par-ticles from it, both for matter and radiation. In general relativity and especially incosmology, the opposite path is taken: one assumes that the properties of matter andradiation are given, e.g. via their equations of state, and one describes in detail thespace-time that follows from them, in particular its curvature. But one fact remainsunchanged throughout all these advances in standard textbook physics: the two con-cepts of particles and of space-time are defined with the help of each other. To avoidthe contradiction between quantum mechanics and general relativity and to try toeliminate their joint incompleteness requires eliminating this circular definition. Asargued in the following, this necessitates a radical change in our description of nature,and in particular about the continuity of space-time.

For a long time, the contradictions in the two descriptions of nature were avoidedby keeping them separate: one often hears the statement that quantum mechanics isvalid at small dimensions, and the other, general relativity, is valid at large dimensions.But this artificial separation is not justified and obviously prevents the solution of theproblem. The situation resembles the well-known drawing by M.C. Escher, where twohands, each holding a pencil, seem to draw each other. If one takes one hand as asymbol for space-time, the other as a symbol for particles, and the act of drawing as a

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symbol for the act of defining, one has a description of standard, present day physics.The apparent contradiction is solved when one recognizes that both concepts (bothhands) result from a hidden third concept (a third hand) from which the other twooriginate. In the picture, this third entity is the hand of the painter.

In the case of space-time and matter, the search for this underlying concept ispresently making renewed progress. [7, 8, 9, 10] The required conceptual changes areso dramatic that they should be of interest to anybody who has an interest in physics.Some of the issues are presented here. The most effective way to study these changesis to focus in detail on that domain where the contradiction between the two standardtheories becomes most dramatic, and where both theories are necessary at the sametime. That domain is given by the following well-known argument.

2 Planck scales

Both general relativity and quantum mechanics are successful theories for the descrip-tion of nature. Each of them provides a criterion to determine when Galilean physicsis not applicable any more. (In the following, we use the terms ‘vacuum’ and ‘emptyspace-time’ interchangeably.)

General relativity shows that it is necessary to take into account the curvature ofspace-time whenever one approaches an object of mass m to distances of the order ofthe Schwarzschild radius rS, given by

rS = 2Gm/c2 . (1)

Approaching the Schwarzschild radius of an object, the difference between generalrelativity and the classical 1/r2 description of gravity becomes larger and larger. Forexample, the barely measurable gravitational deflection of light by the sun is dueto an approach to 2.4 × 105 times the Schwarzschild radius of the sun. [2, 11] Ingeneral however, one is forced to stay away from objects an even larger multiple ofthe Schwarzschild radius, except in the vicinity of neutron stars, as shown in table 1;for this reason, general relativity is not necessary in everyday life. (An object smallerthan its own Schwarzschild radius is called a black hole. Following general relativity,no signals from the inside of the Schwarzschild radius can reach the outside world;hence the name ‘black hole’. [12])

Similarly, quantum mechanics shows that Galilean physics must be abandoned andquantum effects must be taken into account whenever one approaches an object atdistances of the order of the Compton wavelength λC, which is given by

λC =h

mc. (2)

Of course, this length only plays a role if the object itself is smaller than its ownCompton wavelength. At these dimensions one observes relativistic quantum effects,such as particle-antiparticle creation and annihilation. Table 1 shows that the ratiod/λC is near or smaller than 1 only in the microscopic world, so that such effects arenot observed in everyday life; therefore one does not need quantum field theory todescribe common observations.

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Taking these two results together, the situations which require the combined con-cepts of quantum field theory and of general relativity are those in which both con-ditions are satisfied simultaneously. The necessary approach distance is calculated bysetting rS = 2λC (the factor 2 is introduced for simplicity). One finds that this is thecase when lengths or times are of the order of

lPl =√

hG/c3 = 1.6 · 10−35 m, the Planck length,

tPl =√

hG/c5 = 5.4 · 10−44 s , the Planck time.(3)

Whenever one approaches objects at these dimensions, general relativity and quantummechanics both play a role; at these scales effects of quantum gravity appear. Thevalues of the Planck dimensions being extremely small, this level of sophistication isnot necessary in everyday life, neither in astronomy nor in present day microphysics.

However, this sophistication is necessary to understand why the universe is the wayit is. The questions mentioned at the beginning – why do we live in three dimensions,why is the proton 1834 times heavier than the electron – need for their answer a preciseand complete description of nature. The contradictions between quantum mechanicsand general relativity make the search for these answers impossible. On the otherhand, the unified theory, describing quantum gravity, is not yet finished; but a fewglimpses on its implications can already taken at the present stage.

Note that the Planck scales are also the only domain of nature where quantummechanics and general relativity come together; therefore they provide the only possiblestarting point for the following discussion. Planck [13] was interested in the Planckunits mainly as natural units of measurement, and that is the way he called them.However, their importance in nature is much more pervasive, as will be seen now.

3 Farewell to instants of time

The difficulties arising at Planck dimensions appear when one investigates the prop-erties of clocks and meter bars. Is it possible to construct a clock which is able tomeasure time intervals shorter than the Planck time? Surprisingly, the answer is no[14, 15], even though in the relation ∆E∆t ≥ h it seems that by making ∆E arbitrarylarge, one can make ∆t arbitrary small.

Any clock is a device with some moving parts; such parts can be mechanical wheels,matter particles in motion, changing electrodynamic fields – e.g. flying photons –,decaying radioactive particles, etc. For each moving component of a clock, e.g. thetwo hands of the dial, the uncertainty principle applies. As has been discussed inmany occasions [16, 17] and most clearly by Raymer [18], the uncertainty relation fortwo non-commuting variables describes two different, but related situations: it makesa statement about standard deviations of separate measurements on many identicalsystems, and it describes the measurement precision for a joint measurement on asingle system. Throughout this article, only the second viewpoint is used.

Now, in any clock, one needs to know both the time and the energy of each hand,since otherwise it would not be a classical system, i.e. it would not be a recordingdevice. One therefore needs the joint knowledge of non-commuting variables for eachmoving component of the clock; we are interested in the component with the largesttime uncertainty, i.e. imprecision of time measurement ∆t. It is evident that the

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smallest time interval δt which can be measured by a clock is always larger than thetime imprecision ∆t due to the uncertainty relation for its moving components. Thusone has

δt ≥ ∆t ≥h

∆E(4)

where ∆E is the energy uncertainty of the moving component. This energy uncer-tainty ∆E is surely smaller than the total energy E = mc2 of the component itself.(Physically, this condition means that one is sure that there is only one clock; the case∆E > E would mean that it is impossible to distinguish between a single clock, or aclock plus a pair of clock-anticlock, or a component plus two such pairs, etc.) Further-more, any clock provides information; therefore, signals have to be able to leave it. Tomake this possible, the clock may not be a black hole; its mass m must therefore besmaller than the Schwarzschild mass for its size, i.e. m ≤ c2l/G, where l is the size ofthe clock (here and in the following we neglect factors of order unity). Finally, the sizel of the clock must be smaller than c δt itself, to allow a sensible measurement of thetime interval δt: one has l ≤ c δt. (It is amusing to study how a clock larger than c δtstops being efficient, due to the loss of rigidity of its components.) Putting all theseconditions together one after the other, one gets

δt ≥hG

c5δt,

or

δt ≥

√

hG

c5= tPl . (5)

In summary, from three simple properties of every clock, namely that one is sure tohave only one of it, that one can read its dial, and that it gives sensible readouts, onegets the general conclusion that clocks cannot measure time intervals shorter than thePlanck time.

Note that this argument is independent of the nature of the clock mechanism.Whether the clock is powered by gravitational, electrical, plain mechanical or evennuclear means, the relations still apply. Note also that gravitation is essential in thisargument. A well-known study on the limitations of clocks due to their mass and theirmeasuring time has been published by Salecker and Wigner [19] and summarized inpedagogical form by Zimmerman [20]; the present argument differs in that it includesboth quantum mechanics as well as gravity, and therefore yields a different, lower, andmuch more fundamental limit.

The same result can also be found in other ways. [21] For example, any clock smallenough to measure small time intervals necessarily has a certain energy spread, asdescribed by quantum mechanics. Following general relativity, any energy densityinduces a deformation of space-time, and signals from that region arrive with a certaindelay. The energy uncertainty of the source leads to a uncertainty in the delay. Theexpression from general relativity [11] for the deformation of the time part of the lineelement due to a mass m is δt = mG/lc3. If the mass varies by ∆E/c2, the resultinguncertainty ∆t in the delay, the external accuracy of the clock, is

∆t =∆E G

l c5. (6)

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Now the energy uncertainty of the clock is bounded by the uncertainty relation for timeand energy, given the internal time accuracy of the clock. This internal accuracy mustbe smaller or equal than the external one. Putting this all together, one again findsthe relation δt ≤ tPl. One is forced to conclude that in nature there is a minimum timeinterval. In other words, at Planck scales the term “instant of time” has no theoreticalnor experimental backing. It therefore makes no sense to use it.

4 Farewell to points in space

In a similar way one can deduce that it is not possible to make a meter bar or any otherlength measuring device that can measure lengths shorter than the Planck length, asone can find already from lPl = c tPl. [22] The straightforward way to measure thedistance between two points is to put an object at rest at each position. In otherwords, joint measurements of position and momentum are necessary for every lengthmeasurement. Now the minimal length δl that can be measured is surely larger thanthe position uncertainty of the two objects. From the uncertainty principle it is knownthat each object’s position cannot be determined with a precision ∆l smaller than thatgiven by ∆l ∆p = h, where ∆p is the momentum uncertainty. Requiring to have onlyone object at each end means ∆p < mc, which gives

δl ≥ ∆l ≥h

mc. (7)

Furthermore, the measurement cannot be performed if signals cannot leave the object:they may not be black holes. Their masses must therefore be so small that theirSchwarzschild radius rS = 2Gm/c2 is smaller than the distance δl separating them.Dropping again the factor of 2, one gets

δl ≥

√

hG

c3= lPl . (8)

Another way to deduce this limit reverses the role of general relativity and quantumtheory. To measure the distance between two objects, one has to localize the first objectwith respect to the other within a certain interval ∆x. This object thus possesses amomentum uncertainty ∆p ≥ h/∆x and therefore possesses an energy uncertainty∆E = c(c2m2 + (∆p)2)1/2 ≥ ch/∆x. But general relativity shows that a small volumefilled with energy changes the curvature, and thus the metric of the surrounding space.[2, 11] For the resulting distance change ∆l, compared to empty space, one finds[22, 23, 24, 25, 26, 27] the expression ∆l ≈ G∆E/c4. In short, if one localizes a firstparticle in space with a precision ∆x, the distance to a second particle is known onlywith precision ∆l. The minimum length δl that can be measured is obviously largerthan each of the quantities; inserting the expression for ∆E, one finds again that theminimum measurable length δl is given by the Planck length.

As a remark, the Planck length being the shortest possible length, it follows thatthere can be no observations of quantum mechanical effects for situations in which thecorresponding de Broglie or Compton wavelength would be even smaller. This is oneof the reasons why in everyday, macroscopic situations, e.g. in car-car collisions, one

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never observes quantum interference effects, in opposition to the case of proton-protoncollisions.

In summary, from two simple properties common to all length measuring devices,namely that they can be counted and that they can be read out, one arrives at theconclusion that lengths smaller than the Planck length cannot be found in measure-ments. Whatever the method used, whether lengths are measured with a meter bar orby measuring time of flight of particles between the end points: one cannot overcomethis fundamental limit. It follows that in its usual sense as entity without size, theconcept of “point in space” has no experimental backing. In the same way, the term“event”, being a combination of “point in space” and “instant of time”, also loses itsmeaningfulness for the description of nature.

These results are often summarized in the so-called generalized uncertainty principle[22]

∆p∆x ≥ h/2 + fG

c3(∆p)2 , (9)

where f is a numerical factor around unity. A similar expression holds for the time-energy uncertainty relation. The first term on the right hand side is the usual quantummechanical one. The second term, negligible at everyday life energies, plays a role onlynear Planck energies. It is due to the changes in space-time induced by gravity at thesehigh energies. One notes that the generalized principle automatically implies that ∆xcan never be smaller than f1/2lPl.

The generalized uncertainty principle is derived in exactly the same way in whichHeisenberg derived the original uncertainty principle ∆p∆x ≥ h/2 for an object: byusing the deflection of light by the object under a microscope. A careful re-derivationof the process, not disregarding gravity, yields equation (9). [22]

For this reason all approaches which try to unify quantum mechanics and gravitymust yield this relation; indeed it appears in canonical quantum gravity [28], in su-perstring theory [29], and in the quantum group approach [30], sometimes with anadditional term proportional to (∆x)2 on the right hand side of equation (9). [31]

Quantum mechanics starts when one realizes that the classical concept of actionmakes no sense below the value of h; similarly, unified theories such as quantum gravitystart when one realizes that the classical concepts of time and length make no sensenear Planck values. However, the usual description of space-time does contain suchsmall values; it claims the existence of intervals smaller than the smallest measurableone. Therefore the continuum description of space-time has to be abolished in favor ofa more appropriate one.

This is clearly expressed in a new uncertainty relation appearing at Planck scales.[36] Inserting c∆p ≥ ∆E ≥ h/∆t into equation (9) one gets

∆x∆t ≥ hG/c4 = tPllPl , (10)

which of course has no counterpart in standard quantum mechanics. A final wayto convince one-self that points have no meaning is that a point is an entity withvanishing volume; however, the minimum volume possible in nature is the Planckvolume VPl = l3

Pl.

Space-time points are idealizations of events. But this idealization is incorrect. Usingthe concept of “point” is equivalent to the use of the concept of “ether” a century ago:

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one cannot measure it, and it is useful to describe observations only until one has foundthe way to describe them without it.

5 Farewell to the space-time manifold

But the consequences of the Planck limits for time and space measurements can betaken much further. To put the previous results in a different way, points in space andtime have size, namely the Planck size. It is commonplace to say that given any twopoints in space or two instants of time, there is always a third in between. Physicistssloppily call this property continuity, mathematicians call it denseness. However, atPlanck dimensions, this property cannot hold since intervals smaller than the Plancktime can never be found: thus points and instants are not dense, and between two pointsthere is not always a third. But this means that space and time are not continuous.Of course, at large scales they are – approximately – continuous, in the same way thata piece of rubber or a liquid seems continuous at everyday dimensions, but is not atsmall scales. This means that to avoid Zeno’s paradoxes resulting from the infinitedivisibility of space and time one does not need any more the system of differentialcalculus; one can now directly dismiss the paradoxes because of wrong premises on thenature of space and time.

But let us go on. Special relativity, quantum mechanics and general relativity allrely on the idea that time can be defined for all points of a given reference frame.However, two clocks at a distance l cannot be synchronized with arbitrary precision.Since the distance between two clocks cannot be measured with an error smaller thanthe Planck length lPl, and transmission of signals is necessary for synchronization, it isnot possible to synchronize two clocks with a better precision than the time lPl/c = tPl,the Planck time. Due to this impossibility to synchronize clocks precisely, the idea ofa single time coordinate for a whole reference frame is only approximate, and cannotbe maintained in a precise description of nature.

Moreover, since the difference between events cannot be measured with a precisionbetter than a Planck time, for two events distant in time by this order of magnitude,it is not possible to say which one precedes the other with hundred percent certainty.This is an important result. If events cannot be ordered at Planck scales, the conceptof time, which is introduced in physics to describe sequences, cannot be defined at all.In other words, after dropping the idea of a common time coordinate for a completeframe of reference, one is forced to drop the idea of time at a single “point” as well.For example, the concept of ‘proper time’ loses its sense at Planck scales.

In the case of space, it is straightforward to use the same arguments to show thatlength measurements do not allow us to speak of continuous space, but only aboutapproximately continuous space. Due to the lack of measurement precision at Planckscales, the concept of spatial order, of translation invariance and isotropy of the vac-uum, and of global coordinate systems lack experimental backing at those dimensions.

But there is more to come. The very existence of a minimum length contradictsspecial relativity, where it is shown that whenever one changes to a moving coordinatesystem a given length undergoes a Lorentz contraction. A minimum length cannotexist in special relativity; therefore, at Planck dimensions, space-time is neither Lorentzinvariant, nor diffeomorphism invariant, nor dilatation invariant. All the symmetries

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at the basis of special and general relativity are thus only approximately valid at Planckscales.

Due to the imprecision of measurement, most familiar concepts used to describespatial relations become useless. For example, the concept of metric also loses its use-fulness at Planck scales. Since distances cannot be measured with precision, the metriccannot be determined. One deduces that it is impossible to say precisely whether spaceis flat or curved. In other words, the impossibility to measure lengths exactly is equiv-alent to fluctuations of the curvature. [22, 32]

In addition, even the number of space dimensions makes no sense at Planck scales.Let us remind ourselves how one determines this number experimentally. One possi-ble way is to determine how many points one can choose in space such that all theirdistances are equal. If one can find at most n such points, the space has n− 1 dimen-sions. One recognizes directly that without reliable length measurements there is noway to determine reliably the number of dimensions of a space at Planck scales withthis method.

Another simple way to check for three dimensions is to make a knot in a shoestring and glue the ends together: if it stays knotted under all possible deformations,the space has three dimensions. If it can be unknotted, space has more than threedimensions, since in such spaces knots do not exist. Obviously, at Planck dimensionsthe measurement errors do not allow to say whether a string is knotted or not, becauseat crossings one cannot say which strand lies above the other.

There are many other methods to determine the dimensionality of space. [33] Allthese methods use the fact that the concept of dimensionality is based on a precisedefinition of the concept of neighborhood. But at Planck scales, as just mentioned,length measurements do not allow us to say with certainty whether a given pointis inside or outside a given volume. In short, whatever method one uses, the lackof reliable length measurements means that at Planck scales, the dimensionality ofphysical space is not defined. It should therefore not come as a surprise that whenapproaching those scales, one could get a scale-dependent answer, different from three.

The reason for the troubles with space-time become most evident when one remem-bers the well-known definition by Euclid: [34] “A point is that which has no part.” AsEuclid clearly understood, a physical point, and here the stress is on physical, cannotbe defined without some measurement method. A physical point is an idealization ofposition, and as such includes measurement right from the start. In mathematics how-ever, Euclid’s definition is rejected, because mathematical points do not need metricsfor their definition. Mathematical points are elements of a set, usually called a space.In mathematics, a measurable or a metric space is a set of points equipped in addi-tion with a measure or a metric. Mathematical points do not need a metric for theirdefinition; they are basic entities. In contrast to the mathematical situation, the caseof physical space-time, the concepts of measure and of metric are more fundamentalthan that of a point. The difficulty of distinguishing physical and mathematical spacearises from the failure to distinguish a mathematical metric from a physical lengthmeasurement.

Perhaps the most beautiful way to make this point clear is the Banach-Tarski the-orem or paradox, which shows the limits of the concept of volume. [35] This theoremstates that a sphere made of mathematical points can be cut into six pieces in such

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a way that two sets of three pieces can be put together and form two spheres, eachof the same volume as the original one. However, the necessary cuts are “infinitely”curved and thin. For physical matter such as gold, unfortunately – or fortunately –the existence of a minimum length, namely the atomic distance, makes it impossibleto perform such a cut. For vacuum, the puzzle reappears: for example, the energyof its zero-point fluctuations is given by a density times the volume; following theBanach-Tarski theorem, since the concept of volume is ill defined, the total energy isso as well. However, this problem is solved by the Planck length, which provides afundamental length scale also for the vacuum.

In summary, physical space-time cannot be a set of mathematical points. But thesurprises are not finished. At Planck dimensions, since both temporal order and spatialorder break down, there is no way to say if the distance between two near enough space-time regions is space-like or time-like. One cannot distinguish the two cases. At Planckscales, time and space cannot be distinguished from each other. In summary, space-time at Planck scales is not continuous, not ordered, not endowed with a metric, notfour-dimensional, not made of points. If we compare this with the definition of theterm manifold, (a manifold is what locally looks like an euclidean space) not one of itsdefining properties is fulfilled. We arrive at the conclusion that the concept of a space-time manifold has no backing at Planck scales. Even though both general relativityand quantum mechanics use continuous space-time, the combination of both theoriesdoes not. This is one reason why the idea is so slow to disappear.

6 Farewell to observables and measurements

To complete this state of affairs, if space and time are not continuous, all quantitiesdefined as derivatives versus space or time are not defined precisely. Velocity, accel-eration, momentum, energy, etc., are only defined in the classical approximation ofcontinuous space and time. Concepts such as ‘derivative’, ‘divergence-free’, ‘sourcefree’, etc., lose their meaning at Planck scales. Even the important tool of the evo-lution equation, based on derivatives, such as the Schrodinger or the Dirac equation,cannot be used any more.

In fact, all physical observables are defined using at least length and time measure-ments, as is evident from any list of physical units. Any such table shows that allphysical units are products of powers of length, time (and mass) units. (Even thoughin the SI system electrical quantities have a separate base quantity, the ampere, the ar-gument still holds; the ampere is itself defined by measuring a force, which is measuredusing the three base units length, time, and mass.) Now, since time and length are notcontinuous, observables themselves are not continuously varying. This means that atPlanck scales, observables (or their components in a basis) are not to be described byreal numbers with – potentially – infinite precision. Similarly, if time and space are notcontinuous, the usual expression for an observable quantity A, namely A(x, t), doesnot make sense: one has to find a more appropriate description. Physical fields cannotbe described by continuous functions at Planck scales.

In quantum mechanics this means that it makes no sense to define multiplication ofobservables by real numbers. Among others, observables do not form a linear algebra.(One recognizes directly that due to measurement errors, one cannot prove that ob-

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servables do form such an algebra.) This means that observables are not described byoperators at Planck scales. Moreover, the most important observables are the gaugepotentials. Since they do not form an algebra, gauge symmetry is not valid at Planckscales. Even innocuous looking expressions such as [xi, xj ] = 0 for xi 6= xj, which areat the basis of quantum field theory, become meaningless at Planck scales. Even worse,also the superposition principle cannot be backed up by experiment at those scales.

Similarly, permutation symmetry is based on the premise that one can distinguishtwo points by their coordinates, and then exchange particles at those two locations.As just seen, this is not possible if the distance between the two particles is small; oneconcludes that permutation symmetry has no experimental backing at Planck scales.

Even discrete symmetries, like charge conjugation, space inversion, and time reversalcannot be correct in that domain, because there is no way to verify them exactly bymeasurement. CPT symmetry is not valid at Planck scales.

All these results are consistent: if there are no symmetries at Planck scales, therealso are no observables, since physical observables are representations of symmetrygroups. In fact, the limits on time and length measurements imply that the concept ofmeasurement has no significance at Planck scales.

7 Can space-time be a lattice? Can it be dual?

Discretizations of space-time have been studied already fifty years ago. [37] The ideathat space-time is described as a lattice has also been studied in detail, for exampleby Finkelstein [38] and by ’t Hooft. [39] It is generally agreed that in order to get anisotropic and homogeneous situation for large, everyday scales, the lattice cannot beperiodic, but must be random. [40] Moreover any fixed lattice violates the result thatthere are no lengths smaller than the Planck length: due to the Lorentz contraction,any moving observer would find lattice distances smaller than the Planck value.

If space-time is not a set of points or events, it must be a set of something else. Threehints already appear at this stage. The first necessary step to improve the descriptionof motion starts with the recognition that to abandon “points” means to abandon thelocal description of physics. Both quantum mechanics and general relativity assumethat the phrase ‘observable at a point’ had a precise meaning. Due to the impossibilityof describing space as a manifold, this expression is no longer useful. The unificationof general relativity and quantum physics forces a non-local description of nature atPlanck scales.

The existence of a minimal length implies that there is no way to physically distin-guish locations that are spaced by even smaller distances. One is tempted to concludethat therefore any pair of locations cannot be distinguished, even if they are one me-ter apart, since on any path joining two points, any two nearby locations cannot bedistinguished. One notices that this situation is similar to the question on the size of acloud or that of an atom. Measuring water density or electron density, one finds non-vanishing values at any distance from the center; however, an effective size can still bedefined, because it is very improbable to see effects of a cloud’s or of an atom’s presenceat distances much larger than this effective size. Similarly, one guesses that space-timepoints at macroscopic distances can be distinguished because the probability that theywill be confused drops rapidly with increasing distance. In short, one is thus led to

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a probabilistic description of space-time. It becomes a macroscopic observable, thestatistical, or thermodynamic limit of some microscopic entities.

One notes that a fluctuating structure for space-time would also avoid the problemsof fixed structures with Lorentz invariance. This property is of course compatible witha statistical description. In summary, the experimental observations of Lorentz in-variance, isotropy, and homogeneity, together with that of a minimum distance, pointtowards a fluctuating description of space-time. In the meantime, research efforts inquantum gravity, superstring theory and in quantum groups have confirmed indepen-dently from each other that a probabilistic description of space-time, together with anon-local description of observables at Planck dimensions, indeed resolves the contra-dictions between general relativity and quantum theory. To clarify the issue, one hasto turn to the concept of particle.

Before that, a few remarks on one of the most important topics in theoretical physicsat present: duality. String theory is build around a new supposed symmetry space-timeduality (and its generalizations), which states that in nature, physical systems of sizeR are indistinguishable from those of size l2

Pl/R. (In natural units this symmetry is

often written R ↔ 1/R.) Since this relation implies that there is a symmetry betweensystems larger and smaller that the Planck length, it is in contrast with the resultabove that lengths smaller than the Planck length do not make sense. Physical space-time therefore is not dual; but since present string theory includes the assumption thatintervals of any size can be used to describe nature, it needs the duality symmetry tofilter out the thus introduced unphysical situations. Duality does this by explainingthat any system which would have smaller dimensions than the Planck length is infact a system larger than this length.

8 Farewell to particles

Apart from space and time, in every example of motion, there is some object involved.One of the important discoveries of the natural sciences was that all objects are madeof small constituents, called elementary particles. Quantum theory shows that allcomposite, non-elementary objects have a simple property: they have a finite, non-vanishing size. This property allows us to determine whether a particle is elementaryor not. If it behaves like a point particle, it is elementary. At present, only the leptons(electron, muon, tau and the neutrinos), the quarks, and the radiation quanta of theelectromagnetic, the weak and the strong nuclear interaction (the photon, the W andZ bosons, the gluons) have been found to be elementary. [41] A few more elementaryparticles are predicted by various refinements of the standard model. Protons, atoms,molecules, cheese, people, galaxies, etc., are all composite (see table 1). Elementaryparticles are characterized by their vanishing size, their spin, and their mass.

The size of an object, e.g. the one given in table 1, is defined as the length at whichone observes differences from point-like behavior. This is the way in which, using alphaparticle scattering, the radius of the atomic nucleus was determined for the first timein Rutherford’s experiment.

Speaking simply, the size d of an object is determined by measuring the interferencepattern in the scattering of a beam of probe particles. This is the way one determinessizes of objects when one looks at them, using scattered photons. In order to observe

12

such interference effects, the wavelength λ of the probe must be smaller than theobject size d to be determined. The de Broglie wavelength of the probe is given by themass and the relative velocity v between the probe and the unknown object, i.e. oneneeds d > λ = h/(mv) ≥ h/(mc). On the other hand, in order to make a scatteringexperiment possible, the object must not be a black hole, since then it would simplyswallow the infalling particle. This means that its mass m must be smaller than thatof a black hole of its size, i.e., from equation (1), that m < dc2/G; together with theprevious condition one gets

d >

√

hG

c3= lPl . (11)

In other words, there is no way to observe that an object is smaller than the Plancklength. There is thus no way in principle to deduce from observations that a particle ispoint-like. In fact, it makes no sense to use the term “point particle” at all. Of coursethe existence of a minimal length both for empty space and for objects, are related. Ifthe term “point of space” is meaningless, then the term “point particle” is so as well.Note that as in the case of time, the lower limit on length results from the combinationof quantum mechanics and general relativity. (Note also that the minimal size of aparticle has nothing to do with the impossibility, quantum theory, to localize a particleto within better than its Compton wavelength.)

The size d of any elementary particle, which following the conventional quantumfield description is zero, is surely smaller than its own Compton wavelength h/(mc).Moreover, we have seen above that a particle’s size is always larger than the Plancklength: d > lPl. Eliminating the size d one gets a condition for the mass m of anyelementary particle, namely

m <h

c lPl

=

√

hc

G= mPl = 2.2 · 10−8 kg = 1.2 · 1019 GeV/c2 (12)

(This limit, the so-called Planck mass, corresponds roughly to the mass of a ten daysold human embryo, or, equivalently that of a small flea.) In short, the mass of any ele-mentary particle must be smaller than the Planck mass. This fact is already mentionedas “well-known” by Sakharov [42] who explains that these hypothetical particles aresometimes called ‘maximons’. And indeed, the known elementary particles all havemasses well below the Planck mass. (Actually, the question why their masses are soincredibly much smaller than the Planck mass is one of the main questions of highenergy physics. But this is another story.)

There are many other ways to arrive at this mass limit. For example, in order tomeasure mass by scattering – and that is the only way for very small objects – theCompton wavelength of the scatterer must be larger than the Schwarzschild radius;otherwise the probe would be swallowed. Inserting the definition of the two quantitiesand neglecting the factor 2, one gets again the limit m < mPl. (In fact it is a generalproperty of descriptions of nature that a minimum space-time interval leads to anupper limit for elementary particle masses. [43]) The importance of the Planck masswill become clear shortly.

Another property connected with the size of a particle is its electric dipole moment.The stand model of elementary particles gives as upper limit for the dipole moment of

13

the electron de a value of [44]|de| < 10−39 m e (13)

where e is the charge of the electron. This value is ten thousand times smaller thanlPl e; using the fact that the Planck length is the smallest possible length, this implieseither that charge can be distributed in space, or that estimate is wrong, or that thestandard model is wrong, or several of these. Only future will tell.

Let us return to some other strange consequences for particles. In quantum fieldtheory, the difference between a virtual and a real particle is that a real particle ison shell, i.e. it obeys E2 = m2c4 + p2c2, whereas a virtual particle is off shell, i.e.E2 6= m2c4 +p2c2. Due to the fundamental limits of measurement precision, at Planckscales one cannot determine whether a particle is real or virtual.

But that is not all. Since antimatter can be described as matter moving backwards intime, and since the difference between backwards and forwards cannot be determinedat Planck scales, matter and antimatter cannot be distinguished at Planck scales.

Particles are also characterized by their spin. Spin describes two properties of aparticle: its behavior under rotations (and if the particle is charged, the behavior inmagnetic fields) and its behavior under particle exchange. The wave function of spin1 particles remain invariant under rotation of 2π, whereas that of spin 1/2 particleschanges sign. Similarly, the combined wave function of two spin 1 particles does notchange sign under exchange of particles, whereas for two spin 1/2 particles it does.

One sees directly that both transformations are impossible to study at Planck scales.Given the limit on position measurements, the position of the axis of a rotation cannotbe well defined, and rotations become impossible to distinguish from translations.Similarly, position imprecision makes the determination of precise separate positionsfor exchange experiments impossible. In short, spin cannot be defined at Planck scales,and fermions cannot distinguished from bosons, or, differently phrased, matter cannotbe distinguished from radiation at Planack scales. One can thus easily imagine thatsupersymmetry, a unifying symmetry between bosons and fermions, somehow mustbecome important at those dimensions. Let us now move to the main property ofelementary particles.

9 Farewell to mass

The Planck mass divided by the Planck volume, i.e. the Planck density, is given by

ρPl =c5

G2h= 5.2 · 1096 kg/m3 (14)

and is a useful concept in the following. If one wants to measure the (gravitational)mass M enclosed in a sphere of size R and thus (roughly) of volume R3, one way todo this is to put a test particle in orbit around it at a distance R. The universal law ofgravity then gives for the mass M the expression M = Rv2/G, where v is the speed ofthe orbiting test particle. From v < c, one thus deduces that M < c2R/G; using thefact that the minimum value for R is the Planck distance, one gets (neglecting againfactors of order unity) a limit for the mass density, namely

ρ < ρPl . (15)

14

In other words, the Planck density is the maximum possible value for mass density. Inparticular, in a volume of Planck dimensions, one cannot have a larger mass than thePlanck mass.

Interesting things happen when one starts to determine the error ∆M of the massmeasurement in a Planck volume. Let us return to the mass measurement by an or-biting probe. From the relation GM = rv2 one follows G∆M = v2∆r + 2vr∆v >2vr∆v = 2GM∆v/v. For the error ∆v in the velocity measurement one has the un-certainty relation ∆v ≥ h/(m∆r)+ h/(MR) ≥ h/(MR). Inserting this in the previousinequality, and forgetting again the factor of 2, one gets that the mass measurementerror ∆M of a mass M enclosed in a volume of size R follows

∆M ≥h

cR. (16)

Note that for everyday situations, this error is extremely small, and other errors, suchas the technical limits of the balance, are much larger.

As a check, let us take another situation, and use relativistic expressions, in orderto show that the result still holds. Imagine having a mass M in a box of size R andweighing the box. (It is supposed that either the box is massless, or that its mass issubtracted by the scale.) The mass error is given by ∆M = ∆E/c2, where ∆E is dueto the uncertainty in kinetic energy of the mass inside the box. Using the expressionE2 = m2c4 + p2c2 one gets that ∆M ≥ ∆p/c, which again reduces to equation (16).

For a box of Planck dimensions, the mass measurement error is given by the Planckmass. But from above we know that the mass that can be put inside such a box isitself not larger than the Planck mass. In other words, the mass measurement erroris larger (or at best equal) to the mass contained in a box of Planck dimension. Inother words, if one builds a balance with two boxes of Planck size, one empty and theother full, as shown in the figure, nature cannot decide which way the balance shouldhang! Note that even a repeated or a continuous measurement would not resolve thesituation: the balance would then randomly change inclination.

The same surprising answer is found if instead of measuring the gravitational massone measures the inertial mass. The inertial mass for a small object is determinedby touching it, i.e., physically speaking, by performing a scattering experiment. Todetermine the inertial mass inside a region of size R, a probe must have a wavelengthsmaller that R, and thus a corresponding high energy. A high energy means thatthe probe also attracts the particle through gravity. (One thus finds the intermediateresult that at Planck scales, inertial and gravitational mass cannot be distinguished.Also the balance experiment shown in the figure makes this point: at Planck scales,the two effects of mass are always inextricably linked.) In any scattering experiment,e.g. in a Compton-type experiment, the mass measurement is performed by measuringthe wavelength of the probe before and after the scattering experiment. We know fromabove that there always is a minimal wavelength error given by the Planck length lPL.In order to determine the mass in a Planck volume, the probe has to have a wavelengthof the Planck length. In other words, the mass error is as large as the Planck massitself: ∆M ≥ MPl. This limit is thus a direct consequence of the limit on length andspace measurements.

This result has an astonishing consequence. In these examples, the measurementerror is independent of the mass of the scatterer, i.e. independent of whether one

15

starts with a situation in which there is a particle in the original volume, or if there isnone. One thus finds that in a volume of Planck size, it is impossible to say if thereis one particle or none when weighing it or probing it with a beam! In short, vacuum,i.e. empty space-time can not be distinguished from matter at Planck scales. Another,often used way to express this is to say that when a particle of Planck energy travelsthrough space it will be scattered by the fluctuations of space-time itself, making itthus impossible to say whether it was scattered by empty space-time or by matter.These surprising results stem from the fact that whatever definition of mass one uses,it is always measured via length and time measurements. (This is even so for normalweight scales: mass is measured by the displacement of some part of the machine.)And the error in these measurements makes it impossible to distinguish vacuum frommatter.

To put it another way, if one measures the mass of a piece of vacuum of size R, theresult is always at least h/cR; there is no possible way to find a perfect vacuum in anexperiment. On the other hand, if one measures the mass of a particle, one finds thatthe result is size dependent; at Planck dimensions it approaches the Planck mass forevery type of elementary particle. This result is valid both for massive and masslessparticles.

Using another image, when two particles are approached to lengths of the order ofthe Planck length, the error in the length measurements makes it impossible to saywhether there is something or nothing between the two objects. In short, matter andvacuum get mixed-up at Planck dimensions. This is an important result: since bothmass and empty space-time can be mixed-up, one has confirmed that they are madeof the same “fabric”, confirming the idea presented at the beginning. This idea is nowcommonplace in all attempts to find a unified description of nature.

This approach is corroborated by the attempts of quantum mechanics in highlycurved space-time, where a clear distinction between the vacuum and particles is notpossible; the well-known Unruh radiation [45], namely that any observer either accel-erated in vacuum or in a gravitational field in vacuum still detects particles hittinghim, is one of the examples which shows that for curved space-time the idea of vac-uum as a particle-free space does not work. Since at Planck scales it is impossible tosay whether space is flat or not, it follows that it is also impossible to say whether itcontains particles or not.

In summary: the usual concepts of matter and of radiatiation are not applicableat Planck dimensions. Usually, it is assumed that matter and radiation are madeof interacting elementary particles. The concept of an elementary particle is that ofan entity which is countable, point-like, real and not virtual, with a definite mass,a definite spin, distinct from its antiparticle, and most of all, distinct from vacuum,which is assumed to have zero mass. All these properties are found to be incorrect atPlanck scales. One is forced to conclude that at Planck dimensions, it does not makesense to use the concepts of ‘mass’, ‘vacuum’, ‘elementary particle’, ‘radiation’, and‘matter’.

One then can ask at which dimension particles and vacuum can be distinguished.This is obviously possible at everyday dimensions, e.g. of the order of 1 m, and isexperimentally possible up to 10−18 m. Present ideas in particle physics suggest thatparticles are defined at least until the scale of unification of the electromagnetic, weak

16

and strong interaction, i.e. until about 10−31 m. Somewhere between there and thePlanck length of 10−35 m the distinction between particles and space-time becomesimpossible, perhaps in some gradual way.

10 Unification

In this rapid walk, one has thus destroyed all the experimental pillars of quantumtheory: the superposition principle, space-time symmetry, gauge symmetry, and thediscrete symmetries. One also has destroyed the foundations of general relativity,namely the existence of a space-time manifold, of the field concept, the particle concept,and of the concept of mass. It was even shown that matter and space-time cannot bedistinguished. It seems that one has destroyed every concept used for the descriptionof motion, and thus made the description of nature impossible. One naturally askswhether one can save the situation.

To answer this question, one needs to see what one has gained from this sequence ofdestructive arguments. First, since it was found that matter is not distinguishable forvacuum, and since this is correct for all types of particles, be they matter or radiation,one has a clear argument to show that the quest for unification in the description ofelementary particles is correct and necessary.

Moreover, since the concepts ‘mass’, ‘time’, and ‘space’ cannot be distinguishedfrom each other, we also know that a new, single entity is necessary to define bothparticles and space-time. To find out more about this new entity, three approachesare being pursued in present research. The first, quantum gravity, especially the oneusing the Ashtekar’s new variables and the loop representation [7], starts by general-izing space-time symmetry. The second, string theory [8], starts by generalizing gaugesymmetries and interactions, and the third, the algebraic quantum group approach,looks for generalized permutation symmetries. [9]

The basic result of all these descriptions is that what is usually called matter andvacuum are two different aspects of one same “soup” of constituents. In this view,mass and spin are not intrinsic properties, but are properties emerging from certainconfigurations of the fundamental soup. The fundamental constituents themselveshave no mass, no spin, and no charge. The three mentioned approaches to a unifiedtheory have one thing in common: the fundamental constituents are extended. (In thisway the non-locality required above is realized.) In a subsequent article we will give acollection of simple arguments for this result.

The present speculations describe the vacuum as a fluctuating tangle of extendedentities and describe particles as knots or possibly even more complex structures. Suchmodels explain why space has three dimensions, give hope to calculate the masses ofparticles, to determine their spin, and naturally confirm that vacuum has no energyat large scales, but high energy at small dimensions. At everyday energies and lengthscales, the tangle is not visible, the knots become point-like, and one recovers thesmooth, empty vacuum and the massive, spinning, charged and pointlike particles. Inthis way, these speculative models avoid the circular definition of particles and space-time mentioned at the beginning. However, the details of this programme are stillhidden in the impenetrable, but fascinating future of physics.

17

11 Acknowledgement

This work resulted from an intense discussion exchange with Luca Bombelli, now atthe Department of Physics and Astronomy of the University of Mississippi.

References

[*] Present and correspondence address: Philips Research Laboratories, Build-ing WAD1, Prof. Holstlaan 4, 5656 AA Eindhoven, The Netherlands; e-mailschiller@natlab.research.philips.com

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On the other hand, general relativity neglects the commutation rules betweenphysical quantities discovered by experiments on a microscopic scale. General rel-ativity assumes that position and momentum of material objects can be given themeaning of classical physics. General relativity assumes that quantum mechanicsdoes not exist; it ignores Planck’s constant h. Most dramatically, the contradictionis shown by the failure of general relativity to describe the pair creation of spin1/2 particles, a typical quantum process. Wheeler [2, 3] and others [4, 5] haveshown that in such a case, the topology of space necessarily has to change; ingeneral relativity however, the topology of space is fixed. In short, the descriptionof nature of general relativity contradicts that of quantum mechanics, and in orderto describe matter correctly, our concept of space-time has to be changed.

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[13] M. Planck, Uber irreversible Strahlungsvorgange, Sitzungsberichte der Kaiser-lichen Akademie der Wissenschaften zu Berlin, 440–480 1899. Today it is com-monplace to use Dirac’s h instead of Planck’s h, which he originally called b.

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[41] Even though the definition of ‘elementary’ as point particle is all one needs in theargument above, it is not complete, because it seems to leave open the possibilitythat future experiments show that the particles are not elementary. This is not so!In fact any particle smaller than its own Compton wavelength is elementary. If itwere composite, there would be a lighter component inside it; this lighter particlewould have a larger Compton wavelength than the composite particle. This isimpossible, since the size of a composite particle must be larger than the Comptonwavelength of its components. (The possibility that all components be heavierthan the composite, which would avoid this argument, does not seem to lead tosatisfying physical properties; e.g. it leads to intrinsically unstable components.)

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Table

size: mass Schwarz- ratio Compton ratioObject diameter m schild d/rS wave d/λC

d radius rS length λC

galaxy ≈ 1 Zm ≈ 5 · 1040 kg ≈ 70 Tm ≈ 107 ≈ 10−83 m ≈ 10104

neutron star 10 km 2.8 · 1030 kg 4.2 km 2.4 1.3 · 10−73 m 8.0 · 1076

sun 1.4 Gm 2.0 · 1030 kg 3.0 km 4.8 · 105 1.0 · 10−73 m 8.0 · 1081

earth 13 Mm 6.0 · 1024 kg 8.9 mm 1.4 · 109 5.8 · 10−68 m 2.2 · 1074

human 1.8 m 75 kg 0.11 ym 1.6 · 1025 4.7 · 10−45 m 3.8 · 1044

molecule 10 nm 0.57 zg 8.5 · 10−52 m 1.2 · 1043 6.2 · 10−19 m 1.6 · 1010

atom (12C) 0.6 nm 20 yg 3.0 · 10−53 m 2.0 · 1043 1.8 · 10−17 m 3.2 · 107

proton p 2 fm 1.7 yg 2.5 · 10−54 m 8.0 · 1038 2.0 · 10−16 m 9.6pion π 2 fm 0.24 yg 3.6 · 10−55 m 5.6 · 1039 1.5 · 10−15 m 1.4up-quark u < 0.1 fm 0.6 yg 9.0 · 10−55 m < 1.1 · 1038 5.5 · 10−16 m < 0.18electron e < 4 am 9.1 · 10−31 kg 1.4 · 10−57 m 3.0 · 1039 3.8 · 10−13 m < 1.0 · 10−5

neutrino νe < 4 am < 3.0 · 10−35 kg < 4.5 · 10−62 m n.a. > 1.1 · 10−8 m < 3.4 · 10−10

Table 1: The size, Schwarzschild radius, and Compton wavelength of some objectsappearing in nature. A short reminder of the new SI prefixes: f: 10−15, a: 10−18, z: 10−21,y: 10−24, P: 1015, E: 1018, Z: 1021, Y: 1024.

Figure caption

Figure 1: A Gedankenexperiment showing that at Planck scales, matter and vacuumcannot be distinguished.

23

R

M

R